One of the main features that distinguishes modern and early
medieval physics is the mathematical structure of modern physics.
The idea that physics ought to be described in geometrical and mathematical terms
dates back to the ancient Greeks. Whether he was the first to do
so is less clear, but the Greeks strongly associated the idea with
Pythagoras. Plato was particularly influenced by the idea, and his
academy were probably the first group of people to have a serious
attempt at developing a mathematical physics.

They failed.

On the other side of Athens, another man was developing his own
system of philosophy. Aristotle, though a student of Plato, and
similar enough in most aspects of his thought that the Romans
classed him as part of the same school, taught that physics should
be about structure and causality. At the heart of his thought was
the idea that to explain the phenomena of change, a being could
exist in numerous different states (or potentia as he called them).
Change was the excitation of matter from one state to another
(one of the potentia was actualised). Physics becomes, in part,
a description of which transitions are possible, under what
circumstances they are likely to occur, and which transitions occur
naturally on account of the inherent tendencies of matter towards
particular goals. This list of allowed
transitions and interactions allows us to understand the
properties of the beings. Whether something is hard or soft would
depend on its natural tendencies under compression; whether it is
red or blue would depend on its natural interactions with the
colours of light. In particular, each system of beings would have
a natural tendency towards its natural state (or what we would call
its equilibrium state). Bodies made of earth fell because they had
a natural tendency towards a resting point at the centre of the
earth.
Aristotle's physics and astronomy, as refined by the later Greeks,
was immensely successful. It accounted for almost everything.
Therefore it was accepted as being almost certainly true. The
mathematical approach was laid aside.

But not forgotten.

In fourteenth century Oxford and Paris, a group of scholars
wondered whether Pythagoras and Aristotle could be reconciled.
Whether it was possible to develop a mathematical representation of
Aristotle's physics. Physical space, they believed, could be
represented by geometrical space. Events in space could be mapped
to geometrical points. Trajectories of matter could be represented
by geometrical lines, following fixed rules according to the will
of God, regularly directing matter through the secondary causes, its
natural tendencies.

They almost succeeded.

And where they failed was not in their mathematical approach, but
their application of it to Aristotle. Anomalies were found in
Aristotle, places where the physics failed to match up with observation.
Pendulums, the details of projectile motion, the movements of the
planets: it became clear that Aristotle's description of physics
was lacking. So the medieval natural philosophers improved on it.
They introduced the idea of impetus, whose magnitude was a body's
mass times its velocity, and said that it was the impetus of matter
that gradually changed when a body was thrown into the air. Thus they were able to
gradually improve physics, while still remaining Aristotelian in
other areas of their philosophy.

But there was a tension.

It began to look increasingly as though the universe was a great
machine, with matter governed not by internal tendencies but
external laws. The only change that the mathematical approach could
capture was movement from place to place, so that became the only
type of motion accepted. Aristotle's vision of potentia became
redundant, as internal change within substances was seen as
impossible. The fundamental building blocks of matter, corpuscles,
were indestructible, constant, and different arrangements of them
were seen as being sufficient to account for the vast array of
objects around us. God was reduced to being no more than a lawmaker,
and then an idle observer of matter. Thus Aristotle's philosophy was
being challenged by the mathematical mechanical world view. But the
two lived side by side. Mechanism was just the surface picture:
Aristotle still provided the foundation.

The high medieval philosophers almost made the key breakthrough
into modern physics.

But, then, disaster. The black death and wars decimated the
universities. The next generation of scholars did not understand the achievements
of their predecessors. They scorned both the Aristotelian tradition,
and the mathematical approach. Instead they contented themselves
with alchemy and astrology, being content to make observations, plot
them on a graph, and join them up without looking to gain a deeper
understanding. The renaissance was a backwards step in philosophy
and the theoretical understanding of physics. It almost cost
European scholarship everything. It abandoned the possibility of
knowledge of theory.

But it did emphasise and improve the empirical method.

It was an Italian, Galileo Galilei, one of the few of his
generation well trained in the medieval mathematics, who fully realised
that mathematical physics and empirical physics were not enemies, but
allies. He combined them, to devastating effect. Alongside Descartes'
advances in geometry, and Tycho's and Kepler's astronomical observations,
he destroyed the Aristotelian physics once and for all.

And Galileo and Descartes were avowed mechanists, accepting no
compromise with Aristotle.

From Galileo and Kepler, the scientific revolution spread back to
protestant England. Barrow, Boyle, Hooke, and others lead the way;
Newton provided the finishing flourish, both in developing the
mathematics, and applying it. The laws of motion were
established; the mechanical principle governed everything. Gradually
more and more more fields of physics were added to the purview of
mechanical physics. By the end of the nineteenth century, with
mechanical theories of thermal physics, electricity, magnetism and optics
complete, it seemed as though nothing could stand in its way. With
rapid progress in chemistry, and Darwin's theory explaining the
development of life in terms of mechanical principles, nowhere in
science was left untouched by mechanism.

The task of making philosophical sense of the new science was left to the philosophers.

The philosophers were divided. Descartes was there at the start,
but though he believed that the physical world was governed by
mechanical principles, he realised that this could not explain
the mental world, where we give meaning to words and thoughts,
meaning which cannot exist in a purely mechanical system. There
were those who were committed to the empiricist science, as
though Galileo had never occurred, such as Berkeley, Locke and Hume,
who recognised that without Aristotelian natural purposes, there was
no good foundation of ethical thought: no way of getting from a
fact to a purpose and consequently a value in a mechanical system.
Kant tried to bridge the gap between the two, and found himself
questioning how our knowledge related to reality. Marx applied these
theories to social and political sciences, but was blinded by the
idea of class conflict. The social sciences, the theologians, the
educational theorists, they all tried to jump onto the bandwagon of
the success of the mechanical science. The Aristotelian concept of
natural purposes and functions was banished from all corners of the
academy.

And where did this all leave God?

God had moved from being the sustainer of all the universe, to the
sustainer of natural laws, to the creator of natural laws, to an
innocent bystander, to an unnecessary appendage, to something which
didn't exist at all. Western society fell into a practical, if not
yet conceptionally accepted, atheism. The religious, of course, still held
onto their miracles; but the academic elite laughed at them. What
evidence could be strong enough to accept observations and events which
challenged their 'scientific' philosophy?
After all what alternative was there to the
mechanism/dualism/empiricism/Kantian duality/relativism/existentialism/philosophical anarchy
(delete as applicable) which each individual philosopher and natural
scientist believed to be firmly established by science?

But physics had other ideas.

In the early part of the twentieth century, experiments started
showing results that were inconsistent with the mechanical
philosophy. Objects carrying energy and momentum were found to
be neither particles nor waves, but something with some properties
of particles and others of waves. It was found that experimental
results were fundamentally unpredictable, not because because there
were unknown variables in an underlying classical system, but because
there is no underlying classical system. Nature appears to be
fundamentally indeterminate. It is no longer sufficient to specify
a particle's energy and momentum to determine its state, indeed it
is impossible. Instead matter can exist only in a number of discrete
energy levels or states, identified by their momentum, angular
momentum, spin and a few other quantum numbers, which do not include
the particle's location. Only certain
transitions between these states are permitted; some have a natural
tendency to spontaneously occur. The properties of
matter are computed from knowledge of the underlying states. It
is impossible to state all the possible properties of a being
simultaneously; if some are known, then others are wholly undetermined.

A disconnect starts to emerge between modern physics and modern philosophy.

The first attempt at constructing a theory describing these results
was quantum mechanics. The various formulations of quantum mechanics
preserved as much of the mechanical philosophy as they could.
Particles were still seen as being indestructible. Instead of the
particles's position being described by a deterministic
differential equation, now it had a probability amplitude that
evolved deterministically. But, though successful in many respects,
quantum mechanics still did not fully match experiment. So, an
alternative was constructed, quantum field theory, which discarded
the remaining embers of mechanism. There are no differential equations
describing a deterministic evolution of anything. New particles of matter
are continuously created and destroyed. The probabilities of a future
state of matter are computed by computing all possible ways the
universe can evolve from state A to state B,
with each way signified by the different particles being created and
destroyed. There are no forces or potential energies as Galileo and
Newton understood them. The only principle of mechanism that remains is that
space and time can be mapped to a geometrical space. Quantum physics
is thus fundamentally incompatible with enlightenment philosophy.

But it fits in very well with Aristotelian philosophy.

Not perfectly well; Aristotle's philosophy needs to be modified
to fit in with a geometrical view of space and time. So I develop
a precise mathematical definition of form, and show how Aristotelian
causality can be expressed in terms of operators acting on a Hilbert
space. While modern forms of causality are inconsistent with
quantum physics, classical causality is mandated by it. Every aspect
of Aristotle's natural philosophy has an analogue in quantum physics.

So where does this leave God?

The standard objections against God bring in presumptions from
failed enlightenment philosophy, and have no force in classical
philosophy. But the classical arguments for God are given new
strength. Defining God as an uncreatable creator, we can show how
His standard attributes, and His existence, can be carefully
deduced from the underlying metaphysical principles derived from
quantum field theory. In particular, the principles imply a non-mechanical
understanding of physics: that it is a description of how God
sustains the universe. The metaphysical principles lead to a solid
grounding for objective ethical reasoning, contrary to most of the
ways people in modern society tend to ponder morality.

But can we understand physics from knowledge of God?

Quantum field theory can be constructed from a small number of
axioms, among them: indeterminacy, re-normalisability, locality, cluster
decomposition and various local symmetries of the action. Together,
these give the basic mathematical form of field theory, leaving only
some parameters and the numbers of each type of particle to be
determined and measured through observation. If we assume
that physics is a description of God's upholding of the universe,
and that God has the attributes assigned to him by classical theism,
then we can deduce the fundamental axioms of quantum field theory.
If we further assume that God desired to create a universe which
supports complex and rational life, then the parameters which
can't be otherwise computed are constrained to very narrow set of
values - which agree with our measurements.
Thus from accurate knowledge of physics, we can deduce the
existence of God, and from accurate knowledge of God, we can deduce
the form of physical law.

But is more direct evidence for God possible?

A miracle is best defined as a singular event which provides evidence that
God is not indifferent to mankind. The standard arguments against
miracles presuppose a mechanistic understanding of physics. In those
frameworks, a miracle (defined incorrectly as a breaking of the laws of physics)
cannot occur. But in theism, everything is under God's direct
control, without the intermediary of physical law. He
can do what He likes, including special acts of providence to
benefit certain people or mankind as a whole. So the theistic
perspective can account for all the experimental evidence underlying
quantum physics, and also the various pieces of evidence supporting
miraculous events. The mechanist has an alternative model of
physics, which might explain the scientific observations, but
which contradicts the evidence for miraculous events. Since theism
explains all the evidence, and non-theistic models of physics
at best require highly implausible excuses to account for
some of it, clearly theism is to be preferred.

It's just the scientific method of hypothesis, deduction, and
comparison against observation.